3.64 \(\int \left (a+b e^x\right ) \sqrt{c+d x} \, dx\)

Optimal. Leaf size=71 \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]

[Out]

b*E^x*Sqrt[c + d*x] + (2*a*(c + d*x)^(3/2))/(3*d) - (b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqr
t[c + d*x]/Sqrt[d]])/(2*E^(c/d))

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Rubi [A]  time = 0.151445, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^x)*Sqrt[c + d*x],x]

[Out]

b*E^x*Sqrt[c + d*x] + (2*a*(c + d*x)^(3/2))/(3*d) - (b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqr
t[c + d*x]/Sqrt[d]])/(2*E^(c/d))

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Rubi in Sympy [A]  time = 12.4498, size = 61, normalized size = 0.86 \[ \frac{2 a \left (c + d x\right )^{\frac{3}{2}}}{3 d} - \frac{\sqrt{\pi } b \sqrt{d} e^{- \frac{c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{c + d x}}{\sqrt{d}} \right )}}{2} + b \sqrt{c + d x} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*exp(x))*(d*x+c)**(1/2),x)

[Out]

2*a*(c + d*x)**(3/2)/(3*d) - sqrt(pi)*b*sqrt(d)*exp(-c/d)*erfi(sqrt(c + d*x)/sqr
t(d))/2 + b*sqrt(c + d*x)*exp(x)

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Mathematica [A]  time = 0.593539, size = 106, normalized size = 1.49 \[ \sqrt{c+d x} \left (\frac{2 a c}{3 d}+\frac{2 a x}{3}+\frac{b e^{-\frac{c}{d}} \left (-\sqrt{\pi } \text{Erf}\left (\sqrt{-\frac{c+d x}{d}}\right )+2 e^{\frac{c}{d}+x} \sqrt{-\frac{c+d x}{d}}+\sqrt{\pi }\right )}{2 \sqrt{-\frac{c+d x}{d}}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*E^x)*Sqrt[c + d*x],x]

[Out]

Sqrt[c + d*x]*((2*a*c)/(3*d) + (2*a*x)/3 + (b*(Sqrt[Pi] + 2*E^(c/d + x)*Sqrt[-((
c + d*x)/d)] - Sqrt[Pi]*Erf[Sqrt[-((c + d*x)/d)]]))/(2*E^(c/d)*Sqrt[-((c + d*x)/
d)]))

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Maple [A]  time = 0.008, size = 77, normalized size = 1.1 \[ 2\,{\frac{1}{d} \left ( 1/3\, \left ( dx+c \right ) ^{3/2}a+{b \left ( 1/2\,\sqrt{dx+c}{{\rm e}^{{\frac{dx+c}{d}}}}d-1/4\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*exp(x))*(d*x+c)^(1/2),x)

[Out]

2/d*(1/3*(d*x+c)^(3/2)*a+b/exp(c/d)*(1/2*(d*x+c)^(1/2)*exp(1/d*(d*x+c))*d-1/4*d*
Pi^(1/2)/(-1/d)^(1/2)*erf((-1/d)^(1/2)*(d*x+c)^(1/2))))

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Maxima [A]  time = 0.807371, size = 111, normalized size = 1.56 \[ \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a - 3 \,{\left (\frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 2 \, \sqrt{d x + c} d e^{\left (\frac{d x + c}{d} - \frac{c}{d}\right )}\right )} b}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a),x, algorithm="maxima")

[Out]

1/6*(4*(d*x + c)^(3/2)*a - 3*(sqrt(pi)*d*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d)/
sqrt(-1/d) - 2*sqrt(d*x + c)*d*e^((d*x + c)/d - c/d))*b)/d

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Fricas [A]  time = 0.294201, size = 101, normalized size = 1.42 \[ -\frac{3 \, \sqrt{\pi } b d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )} - 2 \,{\left (2 \, a d x + 3 \, b d e^{x} + 2 \, a c\right )} \sqrt{d x + c} \sqrt{-\frac{1}{d}}}{6 \, d \sqrt{-\frac{1}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a),x, algorithm="fricas")

[Out]

-1/6*(3*sqrt(pi)*b*d*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d) - 2*(2*a*d*x + 3*b*d
*e^x + 2*a*c)*sqrt(d*x + c)*sqrt(-1/d))/(d*sqrt(-1/d))

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Sympy [A]  time = 4.42522, size = 76, normalized size = 1.07 \[ \frac{2 a \left (c + d x\right )^{\frac{3}{2}}}{3 d} + b \sqrt{c + d x} e^{- \frac{c}{d}} e^{\frac{c}{d} + x} + \frac{i \sqrt{\pi } b e^{- \frac{c}{d}} \operatorname{erf}{\left (i \sqrt{c + d x} \sqrt{\frac{1}{d}} \right )}}{2 \sqrt{\frac{1}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*exp(x))*(d*x+c)**(1/2),x)

[Out]

2*a*(c + d*x)**(3/2)/(3*d) + b*sqrt(c + d*x)*exp(-c/d)*exp(c/d + x) + I*sqrt(pi)
*b*exp(-c/d)*erf(I*sqrt(c + d*x)*sqrt(1/d))/(2*sqrt(1/d))

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GIAC/XCAS [A]  time = 0.321412, size = 93, normalized size = 1.31 \[ \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a + 3 \,{\left (\frac{\sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-d}} + 2 \, \sqrt{d x + c} d e^{x}\right )} b}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a),x, algorithm="giac")

[Out]

1/6*(4*(d*x + c)^(3/2)*a + 3*(sqrt(pi)*d^2*erf(-sqrt(d*x + c)*sqrt(-d)/d)*e^(-c/
d)/sqrt(-d) + 2*sqrt(d*x + c)*d*e^x)*b)/d